Abstract

An inverse semigroup $S$ is combinatorially factorizable if $S=TG$
where $T$ is a combinatorial (i.e., $\mathcal{H}$ is the equality
relation) inverse subsemigroup of $S$ and $G$ is a subgroup of $S$.
This concept was introduced and studied by Mills, especially in the
case when $S$ is cryptic (i.e., $\mathcal{H}$ is a congruence on
$S$). Her approach is mainly analytical considering subsemigroups of
a cryptic inverse semigroup.
We start with a combinatorial inverse monoid and a factorizable
Clifford monoid and from an action of the former on the latter
construct the semigroups in the title. As a special case, we
consider semigroups which are direct products of a combinatorial
inverse monoid and a group.